Fractional Analysis of Viscous Fluid Flow with Heat and Mass Transfer Over a Flexible Rotating Disk

: An unsteady viscous ﬂuid ﬂow with Dufour and Soret effect, which results in heat and mass transfer due to upward and downward motion of ﬂexible rotating disk, has been studied. The upward or downward motion of non rotating disk results in two dimensional ﬂow, while the vertical action and rotation of the disk results in three dimensional ﬂow. By using an appropriate transformation the governing equations are transformed into the system of ordinary differential equations. The system of ordinary differential equations is further converted into ﬁrst order differential equation by selecting suitable variables. Then, we generalize the model by using the Caputo derivative. The numerical result for the fractional model is presented and validated with Runge Kutta order 4 method for classical case. The compared results are presented in Table and Figures. It is concluded that the fractional model is more realistic than that of classical one, because it simulates the ﬂuid behavior at each fractional value rather than the integral values.


Nomenclature
Greek symbols θ Azimuthal direction η Scaled boundary layer coordinate µ Dynamic viscosity ρ Density k thermal conductivity τ r Radial shear stress on the wall τ θ Azimuthal shear stress on the wall Ω Angular speed of the disk ν = µ ρ Ambient fluid kinematic viscosity C p Specific heat at constant pressure D Molecular diffusion coefficient K T Thermal diffusion ratio σ Fluid electric conductivity µ 0 Free surface permeability κ * Mean absorption coefficient T m Mean fluid temperature σ * Stefan-Boltzmann constant q r = −4σ * 3κ * In the fluid dynamics the problem which has a remarkable application in various field is rotating disk phenomena. Keeping in view its application the research in this field got much attention of researchers in the areas such as, the extraction of energy from hydro power, solar collector and wind energy. The rotating disk phenomena was discussed by many researchers, Von Kármán [Von Kármán (1921)] was the first who studied the flow of fluid on the surface of rotating disk. Cochran [Cochran (1934)] presented more accurate result than that of Von Karman by the help of two series expansion, Benton [Benton (1966)] examined unsteady case and made some modification in Cochran's work. Early, the heat transfer was studied by Millsaps et al. [Millsaps (1951); Riley (1964)] on rotating disk. Kuiken [Kuiken (1971)] clarified blowing effect induced due to rotating porous disk. The influences of rotational deceleration was studied by Watson et al. [Watson and Wang (1979)], they showed a rotating disk in viscous fluid can rotate faster than the disk. The unsteady, laminar flow with decleration over a stretchable rotating disk has studied by Fang et al. [Fang and Tao (2012)]. Türkyılmazoglu et al. [Türkyılmazoglu and Uygun (2004)] derived Karman solution considering compressible flow. Where as surface of the disk partially slips in an infinite viscous fluid was studied by Miklavčič et al. [Miklavčič and Wang (2004)]. Karman's problem was further extended by Turkyilmazoglu [Turkyilmazoglu (2010)]. To examine the magneto hydro dynamics effects, the stretching disk effect is examined by Turkyilmazoglu [Turkyilmazoglu (2012)]. The numerical investigation was presented by Turkyilmazoglu [Turkyilmazoglu (2014)] , by considering the nanoparticle with the rotating disk. The fluid flow due to rotating disk was studied by Hayat et al. [Hayat, Qayyum, Imtiaz et al. (2017)]. The radiation effect and the phenomena of heat transfer has a significant role in space technology. In industries the radiation effect along with high temperature have great applications. Chen et al. [Chen, Armaly and Ali (1984)] considered the thermal radiation in the grey fluid. The study on the rotating disk problem is further extended to hydro magnetic. By using magnetic field [El-Mistikawy (1990)] presented important result. The momentum equation was solved by Rath et al. [Rath and Iyengar (1969)] on porous disk with time dependent flow. The importance and application of fractional calculus (FC) cannot be ignored by someone in advanced technology and in the field of Science and engineering. Fractional calculus characterized those points, at which classical calculus is limited. At the present time fractional calculus is at a stage, where many real world application of science and techologies are still to be introduced. In previous time, FC was considered as esoteric mathematical theory of no applications, for the last decades, due to its wide applications in the field of advanced calculus variation, fractional control of engineering systems, fundamental explorations of the electrical, thermal, mechanical and other properties of several engineering objects like foam, animal tissues, viscoelastic polymers and their scientific importance, understanding of diffusion and wave phenomenas, their verifications and measurments, their role in plasma physics and numerical and analytical tools and techniques, FC has been bringing a tremendous change in research activates. Some researchers have presented several valuable results in monographs and articles [Rath and Iyengar (1969); Ortigueira (2011);Tarasov (2011)]. For special functions and integral transforms FC can be considered as "Laboratory". The applications of FC are mostly due to transcendental functions of the Wright type and Mittag Leffler. The most extensive property of FC in the field of viscoelasticity, in view of its role to model hereditary phenomenas with long memory. Some valuable discussion has been studied about fractional derivative by Benson et al. [Benson, Wheatcraft and Meerschaert (2000)]. Morales-Delgado et al. [Morales-Delgado, Gómez-Aguilar, Saad et al. (2019)] found the solution, using fractional approach, for external forces involving in oxygen diffusion from capillaries to tissue. Gul et al. [Gul, Khan, Noman et al. (2019)] studied the nanofluid passing over a thin needle with fraction order forced convection nanotube. The three dimensional flow of fractional order over rotating disk was studied by Gul et al. [Gul, Khan, Khan et al. (2018)]. Ali et al. [Ali, Jan, Khan et al. (2016)] discussed the solution for Brinkman type fluid with special functions for time free convection flow. Caputo derivative with respect to a kernal function and their application was examined by Almeida et al. [Almeida, Malinowska and Monteiro (2018)]. Turkyilmazoglu [Turkyilmazoglu (2018)] studied fluid flow over single vertically rotating disk with heat transfer. However, no work has been found so far in literature related to single vertically rotating disk with transfer of heat, in which the unsteadiness effect of temperature and concentration is taken. This work is the extension of Turkyilmazoglu's work, in which the unsteadiness effect of concentration and temperature has been taken, which effect the fluid behavior. The fractional behavior of unsteady viscous fluid over a rotating disk has been investigated and presented. The governing equation of fluid flow are numerically investigated by Predictor Corrector method (FDE12) and for comparison Runge Kutta order 4 method is used for classical case. For comparison the result are graphically shown, upto a large extend which give a good agreement with RK4. Next, we formulate the problem with brief discussion.

Mathematical formulation of the problem
Let us consider the impermeable, moving upward/downward rotating disk, which caused the fluid flow as shown in Fig. 1. At t = 0, the location of disk is at a(0) = h, where at some time t the vertical distance is Z = a(t) and the vertical velocity w = a * (t), about z-axis the rotating disk has the angular velocity Ω(t). The velocity components V (u, v, w) are taken along the radial, azimuthal and axial direction respectively. The speed of rotating disk must not be kept constant at any time, otherwise the phenomena will be same as the wall blowing problem, which regarded to a classical rotating disk with a steady fluid flow. The governing equation for the fluid flow motion is as follow [Turkyilmazoglu (2018); Shah, Shuaib and Khan (2017)]: where u, v, w, T, P, and C represent the radial velocity, tangential velocity, axial velocity, Temperature, pressure and concentration respectively. The time dependent temperature is T w (t), while the fluid had ambient temperature T ∞ above the disk. The temperature difference between the ambient and the wall is The imposed boundary condition for infinite rotating disk are as follows: The thermal radiation term is defined as [Shah, Shuaib and Khan (2017)] and η t = −a(t) a(t) (η + 1). (9) The following system of ODE is formed by using Eq. (9) in Eqs.
(1)- (7): The transformed boundary conditions are given by and The physical controlling parameters S of the upward/downward movement of the disk is expressed as: The parameter ω, which represents the constant rotation of the disk is defined as 3 Basics of Caputo derivative Definition 3.1. The fractional integral of order α > 0 for a function g : R + → R is defined by Here and elsewhere Γ denotes the Gamma function.
Definition 3.2. The Caputo fractional order derivative for function g ∈ C n of order α is given below: which is well defined for absolutely continuous functions and n − 1 < α < n ∈ N . Note that the value of the Caputo fractional derivative of the function g at point t involves all the values of g n (χ), for χ ∈ [0, t]. Clearly C D α t (g(t)) tends to g (t) as α → 1.

Results and discussion
In this section, the non-linear system of differential Eqs.          Fig. 13(a) shows the comparison between Runge Kutta order 4 and frictional method F DE12 at α=1.0, both show increasing in temperature θ with enhancement of parameter P r. The frictional solution shows fast convergence than Runge Kutta of order 4, because it can characterize the function at each frictional value of α. Therefore the frictional model show best results. In Fig. 13 Figure 13: Comparison of RK-4 and fractional method taking α = 1 on temperature and concentration profile respectively profile with the variation of thermal radiation parameter Rd has been shown. It is clear from Fig. 13(b) that at initial value of Rd the RK4 show fast convergence. But as we increasing the value greater or equal to 4 the F DE12 change the behavior of fluid concentration and reduces the viscosity of fluid. As a result the concentration increases. Figs. 14(a)-14(e) are plotted for the purpose to reduce error possibilities and get as less error as possible by Caputo derivatives. From observation of Figs. 14(a)-14(e), we can examine that its maximum error is 10 −10 and its minimum error can be reduced up to 10 −4 . That is why error versus different physical parameter's graphs such as S, P r, Rd, Du and Scare plotted. if the findings of this results and figures are used for practical phenomena like engineering purposes, the desired result would be drawn. Tab. 1 shows the behavior of functions such as concentration, temperature and velocities for different values of η and their comparison. We analyzed, the unsteady viscous fluid on combined with heat and mass transfer with Soret and Dufour effect over a impermeable rotating disk, the following results are concluded: • The fractional solution are more fast convergence than that of classical method.
• The increase of axial and tangential velocity and decrease of radial velocity with enhancement of S.
• By the increasing of prandtl number the fluid show anomalous behavior at integer case, but then gradually increase the temperature.
• The enhancement of Dufour parameter the fluid temperature decrease and with Soret So the temperature of fluid increase.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.